On invariant measures of nonlinear Markov processes
type: Item
instance of: scholarly article
title: On invariant measures of nonlinear Markov processes
zbMATH Open document ID: 0806.60046
DOI: 10.1155/S1048953393000310
author: Xinhong Ding
author: N. U. Ahmed
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 1995-02-12T00:00:00Z
full work available at URL: https://eudml.org/doc/47069
review text: The paper deals with the stochastic differential equation in \(R^ d\), \[ dX(t) = \biggl[ - AX(t) + f \bigl( X(t), u(t) \bigr) \biggr] dt + dW(t), \] where \(\mu(t)\) is the probability law of \(X(t)\), \(A\) is a \(d \times d\) matrix, \(f\) is a function from \(R^ d \times M_ 2 (R^ d)\) to \(R^ d\), where \(M_ 2 (R^ d)\) is the space of all probability measures on \(R^ d\) which have finite second moments. The solution \(X(t)\) of this equation is a nonlinear Markov process in the sense of McKean. The aim of the paper is to prove existence and uniqueness of an invariant measure. A probability measure \(\rho\) is said to be invariant if \(\langle \rho, L(\rho) \varphi \rangle = 0\) for all \(C^ \infty\)- functions \(\varphi\) with compact support in \(R^ d\), where \(L(\rho)\) is the infinitesimal generator associated to the coefficients of the differential equation, with \(\mu (t) = \rho\). The authors give preliminary abstract results of existence and uniqueness of invariant measures for general equations. Then they apply these results to the previous particular problem, under appropriate conditions on \(A\) and \(f\).
reviewed by: Franco Flandoli
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 28C10
zbMATH DE Number: 527687
zbMATH Keywords: McKean-Vlasov equation
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: nonlinear Markov process
zbMATH Keywords: existence and uniqueness of an invariant measure
zbMATH Keywords: infinitesimal generator
MaRDI profile type: Publication
OpenAlex ID: W2000254071
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description: scientific article; zbMATH DE number 527687